Along with timeseries measurements, the following static covariates were given for each patient:

• Age and Sex
• GCS (Glasgow Coma Scale) - Integer score from 3 (deep unconsciousness) to 15 (normal)
• Marshall Clasasification - Similar to GCS; Integer from 1 to 6 (lower is better)
• Estimated Initial Time of Injury

As well as an outcome score called "GOS" measured at 3, 6, 12, and 24 months:

• GOS - 1=Dead, 2-3=Severely Disabled, 4-5=Good Outcome

For the sake of modeling, the following assumptions were made:

• Only measurments within 48 hours of initial injury were considered
• Patients with no GCS or Marshall score were excluded
• Patients with less than 8 hours of PbtO2 measurements were also excluded
• Only the 3 month GOS outcome was used, using the 6 month GOS where 3 month not available

There were 339 patients in raw data but only 268 in filtered dataset.

Note: Of 268 patients in filtered set, 18 had missing 3 month GOS

Timeseries measurements for 4 random patients:

Some observations on timeseries values:

• Some quantities are sparse (measured more often for sicker people)
• There are weak correlations between several pairs of variables
• The amount of time between injury and first measurement can vary a lot:

Things should level off by hour 48, validating the assumption that earlier measurements are more relevant.

A big challenge involves working around the fact that this model for the data below is not valid:

Repeating the non-time-dependent variables like this leads to confidence intervals on coefficients that are unrealistically small (and not valid).

Possible Workarounds:

1. Take the mean of each timeseries and use that as a single variable
   This doesn't work – all timeseries variables become insignificant
2. Take multiple distribution statistics (e.g. percentiles) and use those all as variables
   This works better, but has a relatively useless interpretation
3. Try to find "threshold" values where time spent above or below those values become variables
   This is best, though it introduces thresholds as a variable

All of the models that follow take approach 3, though the way they deal with the thresholds will differ.

As a first attempt at modeling, all timeseries variables were separated into ranges based on known guidelines for safe values (Primarily from this Lab Value Guide).

For example, the PaCO2 variable was split into three new variables:

• paco2_0_35 - % of time spent by patient w/ PaCO2 value in [0, 35)
• paco2_35_45 - % of time spent by patient w/ PaCO2 value in [35, 45)
• paco2_45_inf - % of time spent by patient w/ PaCO2 value in [45, +Inf)

For all variables, the known "safe range" was then dropped from the model to avoid linear dependence (percentages across all variables would add up to 1). This leaves only variables indicating time spent at dangerous levels.

A potential improvement on the linear models would involve not needing to set "threshold" ranges for blood gas measurements manually. The most basic version of this kind of model would involve a single threshold and a step function where having blood gas values on one side of that function has a different effect that the other side.

A generalization of this could allow more than one threshold as well as the possibility for gradual shifts between regions of values where effects differ.

A modified logistic model:

\[ logit(y_i) = \alpha + \beta \cdot X_i + f(G_{ij}) \]

where

\[ X_i = [Gender_i, Age_i, CommaScore_i, MarshallScore_i] \]

and

\[ f(G_i) = \frac{1}{n_i} \sum_j{ \frac{c_1}{1 + e^{-c_2(G_{ij} - c_3)}} + \frac{c_4}{1 + e^{-c_5(G_{ij} - c_6)}} } \] \[ n_i = \text{ length of timeseries for patient }i \]

The "double logistic" function definition for \(f\) allows for flexiblity in thresholds and graduated changes.

MCMC sampling model used: Model on Github

These are effect functions drawn from the priors in the model to show possibilities and biases the model begins with: